EDB — 17H

[17H] Topics:separation. Prerequisites:[17D]↺↻.

Given \(A⊂ {\mathbb {R}}^ n\) closed non-empty convex and \(z∉ A\), let \(x^*\) be defined as in the previous exercise [17D]↺↻; define \(𝛿=\| z-x^*\| \), \(v= (z-x^*)/𝛿\) and \(a=⟨ v,x^*⟩\). Prove that \(v,a\) and \(v,a+𝛿\) define two parallel hyperplanes that strongly separate \(z\) from \(A\), in the sense that \(⟨ z,v⟩=a+𝛿\) but \(∀ x∈ A,⟨ x,v⟩≤ a\).